Showing papers on "A priori estimate published in 2018"

The Schauder estimate for kinetic integral equations

Abstract: We establish interior Schauder estimates for kinetic equations with integro-differential diffusion. We study equations of the form $f_t + v \cdot abla_x f = \mathcal L_v f + c$, where $\mathcal L_v$ is an integro-differential diffusion operator of order $2s$ acting in the $v$-variable. Under suitable ellipticity and Holder continuity conditions on the kernel of $\mathcal L_v$, we obtain an a priori estimate for $f$ in a properly scaled Holder space.

Recovering an elastic obstacle containing embedded objects by the acoustic far-field measurements

Abstract: Consider the inverse scattering problem of time-harmonic acoustic waves by a 3D bounded elastic obstacle which may contain embedded impenetrable obstacles inside. We propose a novel and simple technique to show that the elastic obstacle can be uniquely recovered by the acoustic far-field pattern at a fixed frequency, disregarding its contents. Our method is based on constructing a well-posed modified interior transmission problem on a small domain and makes use of an a priori estimate for both the acoustic and elastic wave fields in the usual H 1-norm. In the case when there is no obstacle embedded inside the elastic body, our method gives a much simpler proof for the uniqueness result obtained previously in the literature (Natroshvili et al 2000 Rend. Mat. Serie VII 20 57–92; Monk and Selgas 2009 Inverse Problems Imaging 3 173–98).

Local Existence of MHD Contact Discontinuities

Abstract: We prove the local-in-time existence of solutions with a contact discontinuity of the equations of ideal compressible magnetohydrodynamics (MHD) for two dimensional planar flows provided that the Rayleigh–Taylor sign condition $${[\partial p/\partial N] <0 }$$ on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity. MHD contact discontinuities are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. This paper is a natural completion of our previous analysis (Morando et al. in J Differ Equ 258:2531–2571, 2015) where the well-posedness in Sobolev spaces of the linearized problem was proved under the Rayleigh–Taylor sign condition satisfied at each point of the unperturbed discontinuity. The proof of the resolution of the nonlinear problem given in the present paper follows from a suitable tame a priori estimate in Sobolev spaces for the linearized equations and a Nash–Moser iteration.

Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait

Abstract: We study a mathematical model describing the growth process of a population structured by age and a phenotypical trait, subject to aging, competition between individuals and rare mutations. Our goals are to describe the asymptotic behavior of the solution to a renewal type equation, and then to derive properties that illustrate the adaptive dynamics of such a population. We begin with a simplified model by discarding the effect of mutations, which allows us to introduce the main ideas and state the full result. Then we discuss the general model and its limitations. Our approach uses the eigenelements of a formal limiting operator, that depend on the structuring variables of the model and define an effective fitness. Then we introduce a new method which reduces the convergence proof to entropy estimates rather than estimates on the constrained Hamilton-Jacobi equation. Numerical tests illustrate the theory and show the selection of a fittest trait according to the effective fitness. For the problem with mutations, an unusual Hamiltonian arises with an exponential growth, for which we prove existence of a global viscosity solution, using an uncommon a priori estimate and a new uniqueness result.

Existence of positive solutions for a quasilinear Schrödinger equation

Abstract: This paper is concerned with the quasilinear Schrodinger equation (0.1) − Δ u + V ( x ) u − Δ [ ( 1 + u 2 ) 1 2 ] u 2 ( 1 + u 2 ) 1 2 = μ h ( u ) in R N , where N ≥ 3 and μ > 0 is a parameter. Under some appropriate assumptions on the potential V and the nonlinear term h , we establish the existence of a positive solution for (0.1) with sufficiently large μ . Our method is based on a change of variables, monotonicity trick developed by Jeanjean and a priori estimate.

Strong solutions to a nonlinear stochastic Maxwell equation with a retarded material law

Abstract: We study the Cauchy problem for a semilinear stochastic Maxwell equation with Kerr-type nonlinearity and a retarded material law. We show existence and uniqueness of strong solutions using a refined Faedo–Galerkin method and spectral multiplier theorems for the Hodge–Laplacian. We also make use of a rescaling transformation that reduces the problem to an equation with additive noise to get an appropriate a priori estimate for the solution.

A Discrete Analogue of Energy Integral for a Difference Scheme for Quasilinear Hyperbolic Systems

Abstract: The class of three-dimensional quasilinear hyperbolic systems is studied. The initial boundary value problem for this class of quasilinear hyperbolic systems is given. By constructing the energy’s integral, a priori estimate for the solution of the initial boundary value problem is obtained. Difference scheme is constructed and an a priori estimate for its solution is obtained. Numerical example exhibits the efficiency and accuracy of the method.

Regularity for minimizers of non-autonomous non-quadratic functionals in the case 1 < p < 2: an a priori estimate

Abstract: We prove an a priori estimate for the second derivatives of local minimizers of integral functionals of calculus of variation with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

Construction and research of adequate computational models for quasilinear hyperbolic systems

Abstract: In the paper, we study a class of three-dimensional quasilinear hyperbolic systems. For such system, we set the initial boundary value problem and construct the energy integral. We construct the difference scheme and obtain an a priori estimate for its solution.

Global attractor of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics

Abstract: The objective of this paper is to study the existence of a global attractor in $$(H^2(\Omega ))^3\cap V$$ for the three-dimensional autonomous primitive equations of large-scale ocean and atmosphere dynamics. According to the regularity results for the Stokes-type system in cylinder-type domains established by Ziane (Appl Anal 58(3–4):263–292, 1995), we can only obtain the existence of an absorbing set in $$(H^2(\Omega ))^3\cap V$$ such that the compactness of the semigroup in $$(H^2(\Omega ))^3\cap V$$ cannot be proved by the Sobolev compactness embedding theorem. Therefore, in order to obtain the existence of a global attractor in $$(H^2(\Omega ))^3\cap V,$$ we carry out some a priori estimates of strong solutions to establish the asymptotical compactness of the semigroup in $$(H^2(\Omega ))^3\cap V$$ by asymptotic a priori estimate.

Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space

Abstract: In the present paper, we study a system of viscous conservation laws, which is rewritten to a symmetric hyperbolic-parabolic system, in one-dimensional half space. For this system, we derive a convergence rate of the solutions towards the corresponding stationary solution with/without the stability condition. The essential ingredient in the proof is to obtain the a priori estimate in the weighted Sobolev space. In the case that all characteristic speeds are negative, we show the solution converges to the stationary solution exponentially if an initial perturbation belongs to the exponential weighted Sobolev space. The algebraic convergence is also obtained in the similar way. In the case that one characteristic speed is zero and the other characteristic speeds are negative, we show the algebraic convergence of solution provided that the initial perturbation belongs to the algebraic weighted Sobolev space. The Hardy type inequality with the best possible constant plays an essential role in deriving the optimal upper bound of the convergence rate. Since these results hold without the stability condition, they immediately mean the asymptotic stability of the stationary solution even though the stability condition does not hold.

Navier–Stokes Equation and its Fractional Approximations

Abstract: We consider the Navier–Stokes equation (N-S) in dimensions two and three as limits of the fractional approximations. In 2-D the N-S problem is critical with respect to the standard \(L^2\) a priori estimates and we consider its regular approximations with the fractional power operator \((-P\Delta )^{1+\alpha }\), \(\alpha >0\) small, where P is the projector on the space of divergence-free functions. In 3-D different properties of the N-S problem with respect to the standard \(L^2\) a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator \((-P\Delta )^s\) with \(s>\frac{5}{4}\). Using Dan Henry’s semigroup approach and the Giga-Miyakawa estimates we construct regular solutions to such approximations. The solutions are global in time, unique, smooth and regularized through the equation in time. Solution to 2-D and 3-D N-S equations are obtained next as a limit of such regular solutions of the approximations. Moreover, since the nonlinearity of the N-S equation is of quadratic type, the solutions corresponding to small initial data and small f are shown to be global in time and regular.

Homogenization of a class of singular elliptic problems in perforated domains

Abstract: In this work we study the asymptotic behavior of a class of quasilinear elliptic problems posed in a domain perforated by e -periodic holes of size e . The quasilinear equations present a nonlinear singular lower order term f ζ ( u e ) , where u e is the solution of the problem at e -level, ζ is a continuous function singular in zero and f a function whose summability depends on the growth of ζ near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes contained in Ω and a homogeneous Dirichlet condition on the exterior boundary. The particular case of a Neumann boundary condition on the holes is already new. The main tool in the homogenization process consists in proving a suitable convergence result, which shows that the gradient of u e behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence { u e } , as e → 0 . This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also get a corrector result for our problem. The main novelty of this work is that for the first time the unfolding method is used to treat a singular term as f ζ ( u e ) . This plays an essential role in order to get an almost everywhere convergence of the solution u e , needed in the study the asymptotic behavior of the problem.

Global dynamics of a nonlocal delayed reaction-diffusion equation on a half plane

Abstract: We consider a delayed reaction–diffusion equation with spatial nonlocality on a half plane that describes population dynamics of a two-stage species living in a semi-infinite environment. A Neumann boundary condition is imposed accounting for an isolated domain. To describe the global dynamics, we first establish some a priori estimate for nontrivial solutions after investigating asymptotic properties of the nonlocal delayed effect and the diffusion operator, which enables us to show the permanence of the equation with respect to the compact open topology. We then employ standard dynamical system arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated by the diffusive Nicholson’s blowfly equation and the diffusive Mackey–Glass equation.

Study of Solution for a Parabolic Integrodifferential Equation with the Second Kind Integral Condition

Abstract: In this paper, we establish sufficient conditions for the existence, uniqueness and numerical solution for a parabolic integrodifferential equation with the second kind integral condition. The existence, uniqueness of a strong solution for the linear problem based on a priori estimate “energy inequality” and transformation of the linear problem to linear first-order ordinary differential equation with second member. Then by using a priori estimate and applying an iterative process based on results obtained for the linear problem, we prove the existence, uniqueness of the weak generalized solution of the integrodifferential prob- lem. Also we have developed an efficient numerical scheme, which uses temporary problems with standard boundary conditions. A suitable combination of the auxiliary solutions defines an approximate solution to the original nonlocal problem, the algebraic matrices obtained after the full discretization are tridiagonal, then the solution is obtained by using the Thomas algorithm. Some numerical results are reported to show the efficiency and accuracy of the scheme.

Extension Problems Related to the Higher Order Fractional Laplacian

Abstract: Caffarelli and Silvestre [Comm. Part. Diff. Eqs., 32, 1245–1260 (2007)] characterized the fractional Laplacian (−Δ) s as an operator maps Dirichlet boundary condition to Neumann condition via the harmonic extension problem to the upper half space for 0 0. We also give a new proof to the dissipative a priori estimate of quasi-geostrophic equations in the framework of L p norm using the Caffarelli–Silvestre’s extension technique.

Dirichlet problem of a delay differential equation with spatial non-locality on a half plane

Abstract: In this paper, we are concerned with modeling and analyzing the dynamics for a two-stage species that lives on a half plane. We first derive a spatially nonlocal and temporally delayed differential equation that describes the mature population on a semi-infinite environment with a homogeneous Dirichlet condition. For the derived model, we are able to show that the solutions induce a k -set contraction semiflow with respect to the compact open topology on a bounded positive invariant set attracting every solution of the equation. To describe the global dynamics, we first establish a priori estimate for nontrivial solutions after exploring the delicate asymptotic properties of the nonlocal delayed effect, which enables us to show the repellency of the trivial equilibrium. Using the estimate, k -set contracting property as well as Schauder fixed point theorem, we then establish the existence of a positive spatially heterogeneous steady state. At last, we show global attractivity of the nontrivial steady state by employing dynamical system approaches.

Padé approximants and the modal connection: Towards increased robustness for fast parametric sweeps

Abstract: Summary In order to increase the robustness of a Pade-based approximation of parametric solutions to finite element problems, an a priori estimate of the poles is proposed. The resulting original approach is shown to allow for a straightforward, efficient, subsequent Pade-based expansion of the solution vector components, overcoming some of the current convergence and robustness limitations. In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Pade approximants. The choice of these approximants, as shown in the present work, is theoretically supported by the Montessus de Ballore theorem, concerning the convergence of a series of approximants with fixed denominator degrees. Key features and originality of the proposed approach are i) a component-wise expansion which allows to specifically target subsets of the solution field, and ii) the a priori, simultaneous choice of the Pade approximants and their associated interval of convergence for an effective and more robust approximation. An academic acoustic case study, a structural-acoustic application, and a larger acoustic problem are presented in order to demonstrate the potential of the approach proposed. This article is protected by copyright. All rights reserved.

Random Attractor of Reaction-Diffusion Hopfield Neural Networks Driven by Wiener Processes

Abstract: This paper studies the global existence and uniqueness of the mild solution for reaction-diffusion Hopfield neural networks (RDHNNs) driven by Wiener processes by applying a Schauder fixed point theorem and a priori estimate; then the random attractor for this system is also studied by constructing proper random dynamical system.

On correctors for linear elliptic homogenization in the presence of local defects

Abstract: We consider the corrector equation associated, in homogenization theory , to a linear second-order elliptic equation in divergence form --$\partial$i(aij$\partial$ju) = f , when the diffusion coefficient is a locally perturbed periodic coefficient. The question under study is the existence (and uniqueness) of the corrector, strictly sublinear at infinity, with gradient in L r if the local perturbation is itself L r , r < +$\infty$. The present work follows up on our works [7, 8, 9], providing an alternative, more general and versatile approach , based on an a priori estimate, for this well-posedness result. Equations in non-divergence form such as --aij$\partial$iju = f are also considered, along with various extensions. The case of general advection-diffusion equations --aij$\partial$iju + bj$\partial$ju = f is postponed until our future work [10]. An appendix contains a corrigendum to our earlier publication [9].

Analysis of a time-dependent fluid-solid interaction problem above a local rough surface

Abstract: This paper is concerned with the mathematical analysis of time-dependent fluid-solid interaction problem associated with a bounded elastic body immersed in a homogeneous air or fluid above a local rough surface. We reformulate the unbounded scattering problem into an equivalent initial-boundary value problem defined in a bounded domain by proposing a transparent boundary condition (TBC) on a hemisphere. Analyzing the reduced problem with Lax-Milgram lemma and abstract inversion theorem of Laplace transform, we prove the well-posedness and stability for the reduced problem. Moreover, an a priori estimate is established directly in the time domain for the acoustic wave and elastic displacement with using the energy method.

Existence of weak positive solution for a singular elliptic problem with supercritical nonlinearity

Abstract: In this paper we consider the existence of weak positive solutions for an elliptic problems with the nonlinearity containing both singular and supercritical terms. By means of a priori estimate and sub-and supersolutions method, a positive weak solution is obtained.